Dynamic Billiards

By the Billard ballers

Chaotic Systems History

TODO

Overview of Chaotic Systems: A Descent into Chaos

In order for a Dynamical system to be classified as chaotic it must:

1. Be sensitive to initial conditions
2. Be topologically mixing
3. Have dense periodic orbits

Sensitivity to Initial Conditions

• A driven damped linear oscillator approaches a unique attractor regardless of the initial conditions.
• The difference in solutions Z(x,t), with differing initial conditions, can be expressed as:$$\mid \Delta Z \mid=e^{ƛ t} \mid \Delta Z_0 \mid$$
• As $t \rightarrow \infty$ the transient solutions die out, so ƛ must be negative
• Any Errors in the initial conditions will decay.

Sensitivity to Initial Conditions

• Nonlinear dynamic systems can have many attractors, each of which depends on the initial conditions.
• The difference in solutions of nonlinear systems with differing initial conditions, can also be expressed as:$$\mid \Delta Z \mid=e^{ƛ t} \mid \Delta Z_0 \mid$$
• As $t \rightarrow \infty$ the, ƛ no longer must be negative
• Chaotic systems are all nonlinear

Sensitivity to Initial Conditions

$$\mid \Delta Z \mid=e^{ƛ t} \mid \Delta Z_0 \mid$$

• ƛ is called the Lyapunov exponent
• ƛ is positive in a chaotic system
• So solutions with different initial conditions will diverge exponentially as time passes
• So any errors in the initial conditions will grow exponentially with time. This is what is meant by a sensitivity to initial conditions

Topologically Mixing

Definition: A dynamical system $f$ on a phase space $X$ is said to be topologically mixing if for every two open sets $A$ and $B$ in $X$, there is $N$ sufficiently large such that $f^n(A)\cap B \neq \varnothing$ for all $n \geq N$.

• By Iterating $f$ over any subset of the phase space we can make two different subsets eventually overlap
• This means that any two different trajectories will become arbitrarily regardless of initial conditions

Dense Periodic Orbits

• Any orbits appearing to be periodic are actually arbitrarily close to, but not repeating
• The union of the sets of all these orbits forms the phase space

Physics of the Simulation

• Can be represented as by using a infinite potential well
• Infinite on the boundary and zero inside
• No loss of speed due to friction
• Specular reflection

Can use either Hamiltonian or Lagrangian

Simulation Solutions

• Instead of solving the equations of motion in our simulation we used graphing methods
• Plot a straight line and solve for the intersection with the boundary
• Determine the velocity vectors in the tangent and normal basis
• Flip the sign of the normal velocity component
• repeat

Rectangle Stadium

Periodicity of paths

Rational velocities

import RectTable as rect
from IPython.display import HTML

videoLength=1200

simargs={}
simargs['balls']={0:[0.5,0.5,1.5,0.5]}
simargs['playbackSpeed']=60
simargs['trace']=True
simargs['width']=2
simargs['height']=2
simargs['nBalls']=1
simargs['friction']=False

simulation=rect.RectTable(**simargs)
ani=simulation.main(videoLength)
HTML(ani.to_html5_video())

Rectangle Stadium

Periodicity of Paths

Irrational Velocities

simargs['balls'][0]=[0.5,0.5,1.5*2**(1/2),0.5]
simulation=rect.RectTable(**simargs)
ani=simulation.main(videoLength)
HTML(ani.to_html5_video())

Rectangle Stadium

Sensitivity to initial conditions

simargs['balls']={0:[1,0.7,1.5,0.7],1:[1,0.7,1.51,0.7]}
simargs['nBalls']=2
simulation=rect.RectTable(**simargs)
ani=simulation.main(videoLength)
HTML(ani.to_html5_video())

Rectangle Stadium

Topological Mixing

import random
nballs=30
simargs['trace']=False
simargs['nBalls']=nballs
for i in range(nballs+1):
    
    simargs['balls'][i] = [1.5, 0.75, 1, 0.7]

    # TODO: change this later after we figure out proper initial ball locations
    simargs['balls'][i][2] += random.uniform(-0.1, 0.1)
    simargs['balls'][i][3] += random.uniform(-0.1, 0.1)
    
simulation=rect.RectTable(**simargs)
ani=simulation.main(videoLength)
HTML(ani.to_html5_video())

Buminovich Stadium

Sensitivity to initial conditions

import Buminovich

simargs['balls']={0:[1,0.7,1.5,0.7],1:[1,0.7,1.501,0.7]}
simargs['nBalls']=2
simargs['trace']=True
simulation=Buminovich.Buminovich(**simargs)
ani=simulation.main(videoLength)
HTML(ani.to_html5_video())

Buminovich Stadium

Topological Mixing

nballs=30
simargs['trace']=False
simargs['nBalls']=nballs
for i in range(nballs+1):
    
    simargs['balls'][i] = [1.5, 0.75, 1, 0.7]

    # TODO: change this later after we figure out proper initial ball locations
    simargs['balls'][i][2] += random.uniform(-0.1, 0.1)
    simargs['balls'][i][3] += random.uniform(-0.1, 0.1)
    
simulation=Buminovich.Buminovich(**simargs)
ani=simulation.main(videoLength)
HTML(ani.to_html5_video())

Circle Stadium

Periodicity of Paths

import circle

simargs['balls']={0:[0.5,0.5,1.5,0.5]}
simargs['nBalls']=1
simargs['trace']=True

simulation=circle.CircleTable(**simargs)
ani=simulation.main(videoLength)
HTML(ani.to_html5_video())

Circle Stadium

Sensitivity to initial conditions

simargs['balls']={0:[1,0.7,1.5,0.7],1:[1,0.7,1.51,0.7]}
simargs['nBalls']=2
simargs['trace']=True

simulation=circle.CircleTable(**simargs)
ani=simulation.main(videoLength)
HTML(ani.to_html5_video())

Circle table

Topological Mixing

nballs=30
simargs['trace']=False
simargs['nBalls']=nballs
for i in range(nballs+1):
    
    simargs['balls'][i] = [1.5, 0.75, 1, 0.7]

    # TODO: change this later after we figure out proper initial ball locations
    simargs['balls'][i][2] += random.uniform(-0.1, 0.1)
    simargs['balls'][i][3] += random.uniform(-0.1, 0.1)
    
simulation=circle.CircleTable(**simargs)
ani=simulation.main(videoLength)
HTML(ani.to_html5_video())

Lorentz Gas

The billiard arises from studying the behavior of two interacting disks bouncing inside a square, reflecting off the boundaries of the square and off each other.

Sensitivity to initial conditions

import Lorentz
simargs['balls']={0:[1,0.7,1.5,0.7],1:[1,0.7,1.501,0.7]}
simargs['nBalls']=2
simargs['trace']=True

simulation=Lorentz.Lorentz(**simargs)
ani=simulation.main(videoLength)
HTML(ani.to_html5_video())

Lorentz Gas

Topological Mixing

nballs=30
simargs['trace']=False
simargs['nBalls']=nballs
for i in range(nballs+1):
    
    simargs['balls'][i] = [1.5, 0.75, 1, 0.7]
    simargs['balls'][i][2] += random.uniform(-0.1, 0.1)
    simargs['balls'][i][3] += random.uniform(-0.1, 0.1)
    
simulation=Lorentz.Lorentz(**simargs)
ani=simulation.main(videoLength)
HTML(ani.to_html5_video())